Download CCR High School Math II

MATH II CCR MATH STANDARDS
Mathematical Habits of Mind
1.Makesense ofproblemsandpersevere insolvingthem.
2.Reasonabstractlyandquantitatively.
3.Constructviableargumentsandcritiquethereasoning
ofothers.
4.Modelwithmathematics.
5.Useappropriatetoolsstrategically.
6.Attendtoprecision
7.Lookforandmakeuseofstructure.
8.Lookforandexpress regularityinrepeatedreasoning.
RELATIONSHIPSBETWEENQUANTITIES
Cluster Extendthepropertiesofexponentstorationalexponents.
Explainhowthedefinitionofthemeaningofrationalexponentsfollowsfromextendingthepropertiesof
M.2HS.1 integerexponentstothosevalues,allowingforanotationforradicalsintermsofrationalexponents.(e.g.,
Wedefine51/3tobethecuberootof5becausewewant(5 1/3)3=5(1/3)3 tohold,so(5 1/3)3mustequal5.)
M.2HS.2 Rewriteexpressionsinvolvingradicalsandrationalexponentsusingthepropertiesofexponents.
Cluster Usepropertiesofrationalandirrationalnumbers.
Explainwhysumsandproductsofrationalnumbersarerational,thatthesumofarationalnumberandan
irrationalnumberisirrationalandthattheproductofanonzerorationalnumberandanirrationalnumber
M.2HS.3
isirrational.InstructionalNote:Connecttophysicalsituations,e.g.,findingtheperimeterofasquareof
area2.
Cluster Performarithmeticoperationswithcomplexnumbers.
M.2HS.4
Knowthereisacomplexnumberisuchthati2=−1,andeverycomplexnumberhastheforma+biwitha
andbreal.
Usetherelationi2=–1andthecommutative,associativeanddistributivepropertiestoadd,subtractand
M.2HS.5
multiplycomplexnumbers.InstructionalNote:Limittomultiplicationsthatinvolvei2asthehighestpower
ofi
Cluster Performarithmeticoperationsonpolynomials.
Understandthatpolynomialsformasystemanalogoustotheintegers,namely,theyareclosedunderthe
operationsofaddition,subtraction,andmultiplication;add,subtractandmultiplypolynomials.
M.2HS.6
InstructionalNote:Focusonpolynomialexpressionsthatsimplifytoformsthatarelinearorquadraticina
positiveintegerpowerofx.
QUADRATICFUNCTIONSANDMODELING
Cluster Interpretfunctionsthatariseinapplicationsintermsofacontext.
Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsand
tablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionofthe
relationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionisincreasing,decreasing,
M.2HS.7
positiveornegative;relativemaximumsandminimums;symmetries;endbehavior;andperiodicity.
InstructionalNote:Focusonquadraticfunctions;comparewithlinearandexponentialfunctionsstudied
inMathematicsI.
Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitativerelationshipit
describes.(e.g.,Ifthefunctionh(n)givesthenumberofperson-hoursittakestoassemblenenginesina
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factory,thenthepositiveintegerswouldbeanappropriatedomainforthefunction.)InstructionalNote:
Focusonquadraticfunctions;comparewithlinearandexponentialfunctionsstudiedinMathematicsI.
Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)over
M.2HS.9 aspecifiedinterval.Estimatetherateofchangefromagraph.InstructionalNote:Focusonquadratic
functions;comparewithlinearandexponentialfunctionsstudiedinMathematicsI.
Cluster Analyzefunctionsusingdifferentrepresentations.
Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesand
usingtechnologyformorecomplicatedcases.
a. Graphlinearandquadraticfunctionsandshowintercepts,maxima,andminima.
M.2HS.10
b.
Graphsquareroot,cuberootandpiecewise-definedfunctions,includingstepfunctionsandabsolute
M.2HS.10 valuefunctions.InstructionalNote:Compareandcontrastabsolutevalue,stepandpiecewisedefined
M.2HS.11
functionswithlinear,quadratic,andexponentialfunctions.Highlightissuesofdomain,rangeand
usefulnesswhenexaminingpiecewise-definedfunctions.
InstructionalNote:Extendworkwithquadraticstoincludetherelationshipbetweencoefficientsandroots
andthatoncerootsareknown,aquadraticequationcanbefactored.
Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealandexplaindifferent
propertiesofthefunction.
a. Usetheprocessoffactoringandcompletingthesquareinaquadraticfunctiontoshowzeros,
extremevaluesandsymmetryofthegraphandinterprettheseintermsofacontext.
b. Usethepropertiesofexponentstointerpretexpressionsforexponentialfunctions.(e.g.,Identify
percentrateofchangeinfunctionssuchasy=(1.02)t,y=(0.97)t,y=(1.01) 12t,y=(1.2) t/10,andclassify
themasrepresentingexponentialgrowthordecay.)InstructionalNote:Thisunitand,inparticular,this
standardextendstheworkbeguninMathematicsIonexponentialfunctionswithintegerexponents.
InstructionalNote:Extendworkwithquadraticstoincludetherelationshipbetweencoefficientsandroots
andthatoncerootsareknown,aquadraticequationcanbefactored.
Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,
numericallyintables,orbyverbaldescriptions).(e.g.,Givenagraphofonequadraticfunctionandan
algebraicexpressionforanother,saywhichhasthelargermaximum).InstructionalNote:Focuson
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expandingthetypesoffunctionsconsideredtoinclude,linear,exponentialandquadratic.Extendwork
withquadraticstoincludetherelationshipbetweencoefficientsandrootsandthatoncerootsareknown,
aquadraticequationcanbefactored.
Cluster Buildafunctionthatmodelsarelationshipbetweentwoquantities.
Writeafunctionthatdescribesarelationshipbetweentwoquantities.
a. Determineanexplicitexpression,arecursiveprocessorstepsforcalculationfromacontext.
M.2HS.13 b.Combinestandardfunctiontypesusingarithmeticoperations.(e.g.,Buildafunctionthatmodelsthe
temperatureofacoolingbodybyaddingaconstantfunctiontoadecayingexponential,andrelatethese
functionstothemodel.InstructionalNote:Focusonsituationsthatexhibitaquadraticorexponential
relationship.
Cluster Buildnewfunctionsfromexistingfunctions.
Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk
(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experimentwithcasesandillustratean
M.2HS.14 explanationoftheeffectsonthegraphusingtechnology.Includerecognizingevenandoddfunctionsfrom
theirgraphsandalgebraicexpressionsforthem.InstructionalNote:Focusonquadraticfunctionsand
considerincludingabsolutevaluefunctions.
Findinversefunctions.Solveanequationoftheformf(x)=cforasimplefunctionfthathasaninverse
M.2HS.15
andwriteanexpressionfortheinverse.Forexample,f(x)=2x3orf(x)=(x+1)/(x-1)forx≠1.Instructional
Note:Focusonlinearfunctionsbutconsidersimplesituationswherethedomainofthefunctionmustbe
restrictedinorderfortheinversetoexist,suchasf(x)=x2 ,x>0.
Cluster Constructandcomparelinear,quadratic,andexponentialmodelsandsolveproblems.
Usinggraphsandtables,observethataquantityincreasingexponentiallyeventuallyexceedsaquantity
M.2HS.16 increasinglinearly,quadratically;or(moregenerally)asapolynomialfunction.InstructionalNote:
ComparelinearandexponentialgrowthstudiedinMathematicsItoquadraticgrowth.
EXPRESSIONSANDEQUATIONS
Cluster Interpretthestructureofexpressions.
Interpretexpressionsthatrepresentaquantityintermsofitscontext.
a.
b.
Interpretpartsofanexpression,suchasterms,factors,andcoefficients.
Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.For
M.2HS.17 example,interpretP(1+r)n astheproductofPandafactornotdependingonP.
M.2HS.17
InstructionalNote:Focusonquadraticandexponentialexpressions.Exponentsareextendedfromthe
integerexponentsfoundinMathematicsItorationalexponentsfocusingonthosethatrepresentsquare
orcuberoots.
Usethestructureofanexpressiontoidentifywaystorewriteit.Forexample,seex4–y4as(x2)2–(y2)2,thus
M.2HS.18 recognizingitasadifferenceofsquaresthatcanbefactoredas(x2–y2)(x2+y2).InstructionalNote:Focus
onquadraticandexponentialexpressions.
Cluster Writeexpressionsinequivalentformstosolveproblems.
Chooseandproduceanequivalentformofanexpressiontorevealandexplainpropertiesofthequantity
representedbytheexpression.
a. Factoraquadraticexpressiontorevealthezerosofthefunctionitdefines.
b.Completethesquareinaquadraticexpressiontorevealthemaximumorminimumvalueofthe
functionitdefines.
M.2HS.19 c.Usethepropertiesofexponentstotransformexpressionsforexponentialfunctions.Forexamplethe
expression1.15tcanberewrittenas(1.151/12)12t≈1.01212ttorevealtheapproximateequivalentmonthly
interestrateiftheannualrateis15%.
InstructionalNote:Itisimportanttobalanceconceptualunderstandingandproceduralfluencyinwork
withequivalentexpressions.Forexample,developmentofskillinfactoringandcompletingthesquare
goeshand-in-handwithunderstandingwhatdifferentformsofaquadraticexpressionreveal.
Cluster Createequationsthatdescribenumbersorrelationships.
Createequationsandinequalitiesinonevariableandusethemtosolveproblems.InstructionalNote:
M.2HS.20 Includeequationsarisingfromlinearandquadraticfunctions,andsimplerationalandexponential
functions.ExtendworkonlinearandexponentialequationsinMathematicsItoquadraticequations.
Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graphequations
M.2HS.21 oncoordinateaxeswithlabelsandscales.InstructionalNote:Extendworkonlinearandexponential
equationsinMathematicsItoquadraticequations.
Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolvingequations.
(e.g.,RearrangeOhm’slawV=IRtohighlightresistanceR.)InstructionalNote:Extendtoformulas
M.2HS.22
involvingsquaredvariables.ExtendworkonlinearandexponentialequationsinMathematicsIto
quadraticequations.
Cluster Solveequationsandinequalitiesinonevariable.
Solvequadraticequationsinonevariable.
a.Usethemethodofcompletingthesquaretotransformanyquadraticequationinxintoanequationof
theform(x–p)2=qthathasthesamesolutions.Derivethequadraticformulafromthisform.
2
M.2HS.23 b.Solvequadraticequationsbyinspection(e.g.,forx =49),takingsquareroots,completingthesquare,
thequadraticformulaandfactoring,asappropriatetotheinitialformoftheequation.Recognizewhenthe
quadraticformulagivescomplexsolutionsandwritethemasa±biforrealnumbersaandb.
InstructionalNote:Extendtosolvinganyquadraticequationwithrealcoefficients,includingthosewith
complexsolutions.
Cluster Usecomplexnumbersinpolynomialidentitiesandequations.
M.2HS.24
Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions.InstructionalNote:Limitto
quadraticswithrealcoefficients.
2
M.2HS.25 Extendpolynomialidentitiestothecomplexnumbers.Forexample,rewritex +4as(x+2i)(x–2i).
(+)
InstructionalNote:Limittoquadraticswithrealcoefficients.
M.2HS.26 KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.Instructional
(+)
Note:Limittoquadraticswithrealcoefficients.
Cluster Solvesystemsofequations.
Solveasimplesystemconsistingofalinearequationandaquadraticequationintwovariables
algebraicallyandgraphically.(e.g.,Findthepointsofintersectionbetweentheliney=–3xandthecirclex2
2
M.2HS.27 +y =3.)InstructionalNote:Includesystemsthatleadtoworkwithfractions.(e.g.,Findingthe
intersectionsbetweenx2+y2=1andy=(x+1)/2leadstothepoint(3/5,4/5)ontheunitcircle,
correspondingtothePythagoreantriple32 +42=52.)
APPLICATIONSOFPROBABILITY
Cluster Understandindependenceandconditionalprobabilityandusethemtointerpretdata.
M.2HS.28
M.2HS.29
M.2HS.30
M.2HS.31
M.2HS.32
Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristics(orcategories)of
theoutcomesorasunions,intersectionsorcomplementsofotherevents(“or,”“and,”“not”).
UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurringtogetheristhe
productoftheirprobabilitiesandusethischaracterizationtodetermineiftheyareindependent.
UnderstandtheconditionalprobabilityofAgivenBasP(AandB)/P(B),andinterpretindependenceofA
andBassayingthattheconditionalprobabilityofAgivenBisthesameastheprobabilityofA,andthe
conditionalprobabilityofBgivenAisthesameastheprobabilityofB.
Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesareassociatedwitheach
objectbeingclassified.Usethetwo-waytableasasamplespacetodecideifeventsareindependentand
toapproximateconditionalprobabilities.(e.g.,Collectdatafromarandomsampleofstudentsinyour
schoolontheirfavoritesubjectamongmath,scienceandEnglish.Estimatetheprobabilitythatarandomly
selectedstudentfromyourschoolwillfavorsciencegiventhatthestudentisintenthgrade.Dothesame
forothersubjectsandcomparetheresults.)InstructionalNote:Buildonworkwithtwo-waytablesfrom
MathematicsItodevelopunderstandingofconditionalprobabilityandindependence.
Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageand
everydaysituations.(e.g.,Comparethechanceofhavinglungcancerifyouareasmokerwiththechance
ofbeingasmokerifyouhavelungcancer.)
Cluster Usetherulesofprobabilitytocomputeprobabilitiesofcompoundeventsinauniformprobabilitymodel.
FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomesthatalsobelongtoAand
interprettheanswerintermsofthemodel.
ApplytheAdditionRule,P(AorB)=P(A)+P(B)–P(AandB),andinterprettheanswerintermsofthe
M.2HS.34
model.
M.2HS.35 ApplythegeneralMultiplicationRuleinauniformprobabilitymodel,P(AandB)=P(A)P(B|A)=P(B)P(A|B),
(+)
andinterprettheanswerintermsofthemodel.
M.2HS.33
M.2HS.36
Usepermutationsandcombinationstocomputeprobabilitiesofcompoundeventsandsolveproblems.
(+)
Cluster Useprobabilitytoevaluateoutcomesofdecisions.
M.2HS.37
Useprobabilitiestomakefairdecisions(e.g.,drawingbylotsorusingarandomnumbergenerator).
(+)
Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.,producttesting,medicaltesting,and/or
pullingahockeygoalieattheendofagame).InstructionalNote:Thisunitsetsthestageforworkin
M.2HS.38
MathematicsIII,wheretheideasofstatisticalinferenceareintroduced.Evaluatingtherisksassociated
(+)
withconclusionsdrawnfromsampledata(i.e.,incompleteinformation)requiresanunderstandingof
probabilityconcepts.
SIMILARITY,RIGHTTRIANGLETRIGONOMETRY,ANDPROOF
Cluster Understandsimilarityintermsofsimilaritytransformations
Verifyexperimentallythepropertiesofdilationsgivenbyacenterandascalefactor.
a.Adilationtakesalinenotpassingthroughthecenterofthedilationtoaparallellineandleavesaline
M.2HS.39
passingthroughthecenterunchanged.
b.Thedilationofalinesegmentislongerorshorterintheratiogivenbythescalefactor.
Giventwofigures,usethedefinitionofsimilarityintermsofsimilaritytransformationstodecideiftheyare
M.2HS.40 similar;explainusingsimilaritytransformationsthemeaningofsimilarityfortrianglesastheequalityofall
correspondingpairsofanglesandtheproportionalityofallcorrespondingpairsofsides.
M.2HS.41 UsethepropertiesofsimilaritytransformationstoestablishtheAAcriterionfortwotrianglestobesimilar.
Cluster Provegeometrictheorems.
Provetheoremsaboutlinesandangles.Theoremsinclude:verticalanglesarecongruent;whena
transversalcrossesparallellines,alternateinterioranglesarecongruentandcorrespondinganglesare
congruent;pointsonaperpendicularbisectorofalinesegmentareexactlythoseequidistantfromthe
segment’sendpoints.Implementationmaybeextendedtoincludeconcurrenceofperpendicularbisectors
M.2HS.42
andanglebisectorsaspreparationforM.2HS.C.3.InstructionalNote:Encouragemultiplewaysofwriting
proofs,suchasinnarrativeparagraphs,usingflowdiagrams,intwo-columnformat,andusingdiagrams
withoutwords.Studentsshouldbeencouragedtofocusonthevalidityoftheunderlyingreasoningwhile
exploringavarietyofformatsforexpressingthatreasoning.
Provetheoremsabouttriangles.Theoremsinclude:measuresofinterioranglesofatrianglesumto180°;
baseanglesofisoscelestrianglesarecongruent;thesegmentjoiningmidpointsoftwosidesofatriangleis
paralleltothethirdsideandhalfthelength;themediansofatrianglemeetatapoint.InstructionalNote:
Encouragemultiplewaysofwritingproofs,suchasinnarrativeparagraphs,usingflowdiagrams,intwoM.2HS.43
columnformat,andusingdiagramswithoutwords.Studentsshouldbeencouragedtofocusonthevalidity
oftheunderlyingreasoningwhileexploringavarietyofformatsforexpressingthatreasoning.
Implementationofthisstandardmaybeextendedtoincludeconcurrenceofperpendicularbisectorsand
anglebisectorsinpreparationfortheunitonCirclesWithandWithoutCoordinates.
Provetheoremsaboutparallelograms.Theoremsinclude:oppositesidesarecongruent,oppositeangles
arecongruent,thediagonalsofaparallelogrambisecteachotherandconversely,rectanglesare
parallelogramswithcongruentdiagonals.InstructionalNote:Encouragemultiplewaysofwritingproofs,
M.2HS.44
suchasinnarrativeparagraphs,usingflowdiagrams,intwo-columnformatandusingdiagramswithout
words.Studentsshouldbeencouragedtofocusonthevalidityoftheunderlyingreasoningwhileexploring
avarietyofformatsforexpressingthatreasoning.
Cluster Provetheoremsinvolvingsimilarity.
Provetheoremsabouttriangles.Theoremsinclude:alineparalleltoonesideofatriangledividestheother
twoproportionallyandconversely;thePythagoreanTheoremprovedusingtrianglesimilarity.
Usecongruenceandsimilaritycriteriafortrianglestosolveproblemsandtoproverelationshipsin
M.2HS.46
geometricfigures.
M.2HS.45
Cluster Usecoordinatestoprovesimplegeometrictheoremsalgebraically.
M.2HS.47
Findthepointonadirectedlinesegmentbetweentwogivenpointsthatpartitionsthesegmentinagiven
ratio.
Cluster Definetrigonometricratiosandsolveproblemsinvolvingrighttriangles.
M.2HS.48
Understandthatbysimilarity,sideratiosinrighttrianglesarepropertiesoftheanglesinthetriangle,
leadingtodefinitionsoftrigonometricratiosforacuteangles.
M.2HS.49 Explainandusetherelationshipbetweenthesineandcosineofcomplementaryangles.
M.2HS.50 UsetrigonometricratiosandthePythagoreanTheoremtosolverighttrianglesinappliedproblems.
Cluster Proveandapplytrigonometricidentities.
ProvethePythagoreanidentitysin2(θ)+cos2(θ)=1anduseittofindsin(θ),cos(θ),ortan(θ),givensin(θ),
cos(θ),ortan(θ),andthequadrantoftheangle.InstructionalNote:Limitθtoanglesbetween0and90
M.2HS.51
degrees.ConnectwiththePythagoreantheoremandthedistanceformula.Extensionoftrigonometric
functionstootheranglesthroughtheunitcircleisincludedinMathematicsIII.
CIRCLESWITHANDWITHOUTCOORDINATES
Cluster Understandandapplytheoremsaboutcircles.
M.2HS.52 Provethatallcirclesaresimilar.
Identifyanddescriberelationshipsamonginscribedangles,radiiandchords.Includetherelationship
M.2HS.53 betweencentral,inscribedandcircumscribedangles;inscribedanglesonadiameterarerightangles;the
radiusofacircleisperpendiculartothetangentwheretheradiusintersectsthecircle.
Constructtheinscribedandcircumscribedcirclesofatriangleandprovepropertiesofanglesfora
M.2HS.54
quadrilateralinscribedinacircle.
M.2HS.55
Constructatangentlinefromapointoutsideagivencircletothecircle.
(+)
Cluster Findarclengthsandareasofsectorsofcircles.
Deriveusingsimilaritythefactthatthelengthofthearcinterceptedbyanangleisproportionaltothe
radiusanddefinetheradianmeasureoftheangleastheconstantofproportionality;derivetheformula
fortheareaofasector.InstructionalNote:Emphasizethesimilarityofallcircles.Notethatbysimilarity
M.2HS.56
ofsectorswiththesamecentralangle,arclengthsareproportionaltotheradius.Usethisasabasisfor
introducingradianasaunitofmeasure.Itisnotintendedthatitbeappliedtothedevelopmentofcircular
trigonometryinthiscourse.
Cluster Translatebetweenthegeometricdescriptionandtheequationforaconicsection.
DerivetheequationofacircleofgivencenterandradiususingthePythagoreanTheorem;completethe
M.2HS.57 squaretofindthecenterandradiusofacirclegivenbyanequation.InstructionalNote:Connectthe
equationsofcirclesandparabolastopriorworkwithquadraticequations.
Derivetheequationofaparabolagiventhefocusanddirectrix.InstructionalNote:Thedirectrixshouldbe
M.2HS.58
paralleltoacoordinateaxis.
Cluster Usecoordinatestoprovesimplegeometrictheoremsalgebraically.
Usecoordinatestoprovesimplegeometrictheoremsalgebraically.(e.g.,Proveordisprovethatafigure
definedbyfourgivenpointsinthecoordinateplaneisarectangle;proveordisprovethatthepoint(1,√3)
M.2HS.59
liesonthecirclecenteredattheoriginandcontainingthepoint(0,2).)InstructionalNote:Includesimple
proofsinvolvingcircles.
Cluster Explainvolumeformulasandusethemtosolveproblems.
Giveaninformalargumentfortheformulasforthecircumferenceofacircle,areaofacircle,volumeofa
cylinder,pyramid,andcone.Usedissectionarguments,Cavalieri’sprincipleandinformallimitarguments.
M.2HS.60 InstructionalNote:Informalargumentsforareaandvolumeformulascanmakeuseofthewayinwhich
areaandvolumescaleundersimilaritytransformations:whenonefigureintheplaneresultsfromanother
M.2HS.61
byapplyingasimilaritytransformationwithscalefactork,itsareaisk2timestheareaofthefirst.
Usevolumeformulasforcylinders,pyramids,conesandspherestosolveproblems.Volumesofsolid
figuresscalebyk3underasimilaritytransformationwithscalefactork.